Unimodular Diffusion and Interacting Vacuum Cosmology

We investigate the correspondence between unimodular diffusion cosmology and interacting dark sector models at the background and linear perturbation levels. In the diffusion framework, the effective cosmological constant becomes time dependent, $\mathrm{\Lambda }(t)$, sourced by a diffusion current. We show that at the background level this framework can be mapped onto interacting dark energy models with $w=-1$ and energy transfer $Q$. Using two common parameterizations, $Q=\xi H{\rho }_{\mathrm{d}\mathrm{e}}$ and $Q=\xi H{\rho }_{\mathrm{d}\mathrm{m}}$, and data from supernovae, DESI BAO, cosmic chronometers, and CMB distance priors, we find $\xi =-0.0197\pm 0.0076$ for the vacuum-coupled case, while the matter-coupled case gives a best-fit $\xi =0.0018$ with comparable goodness of fit. At the level of linear perturbations, however, the diffusion framework is consistent only with interacting vacuum models having homogeneous energy transfer ($Q\propto {\rho }_{\mathrm{d}\mathrm{e}}$ with $\delta Q=0$), thereby breaking the degeneracy with more general interacting dark energy scenarios. Including redshift-space distortion data, we obtain $\xi =-0.0147\pm 0.0075$, consistent with $\mathrm{\Lambda }$CDM ($\xi =0$) at $2\sigma$. The inferred clustering amplitude is $S_8=0.782\pm 0.026$ for the diffusion model, compared to $S_8=0.77\pm 0.025$ for $\mathrm{\Lambda }$CDM under the same dataset, indicating a modest but non-negligible impact on structure growth.